a. Start with the model with 7 parameters and through parameter rescaling arrive to 4-parameter model
b. In the presented version, protease production was assumed constant. How do the results change if you include the fact that protease is induced by
i. Proto-fibrils (s)
ii. Oligomers (ααα)
iii. Fibrils, (F)
c. Which of i-iii have a maximal impact on model outcome?
d. How would you modify the model to include the effect of autophagy? How does this change the results?
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve_original_model(gamma=1,m=25,sigma=1,nu=1,
ic = [0,0,0], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
sol_m25 = solve_original_model(m=25,tmax= 250)
sol_m10 = solve_original_model(m=10,tmax=220)
plt.plot(sol_m25[0], sol_m25[1],label="F(t)",color="blue")
plt.plot(sol_m25[0],sol_m25[2],label="C(t)",color="green")
plt.plot(sol_m25[0],sol_m25[3],label="P(t)",color="red")
plt.xlim([50,220])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with m=25, $\nu = \gamma = \sigma =1$")
plt.show()
plt.plot(sol_m10[0], sol_m10[1],label="F(t)",color="blue")
plt.plot(sol_m10[0],sol_m10[2],label="P(t)",color="red")
plt.plot(sol_m10[0],sol_m10[3],label="C(t)",color="green")
plt.xlim([50,220])
plt.ylim([0,100])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with m=10, $\nu = \gamma = \sigma =1$")
plt.show()
def num_oligomers(sol):
ts = sol[0]
Ps = sol[3]
cumulative_integral = [0] * len(ts)
for i in range(2, len(ts)):
area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
cumulative_integral[i] = cumulative_integral[i-1] + area
return([ts,cumulative_integral])
# oligomers_m25 = num_oligomers(sol_m25)
oligomers_m10 = num_oligomers(sol_m10)
plt.plot(oligomers_m10[0],oligomers_m10[1],color="red")
plt.title(r"Oligomer accumultion $\left(\int 1/P dt\right)$ for $m=10$")
plt.xlabel("Time")
plt.ylabel("Oligomer accumulation")
plt.ylim([0,700])
plt.xlim([50,220])
# plt.plot(oligomers_m10[0],oligomers_m10[1])
(50.0, 220.0)
solve_original_model(m=1, sigma=1,tmax=2000,start_from=50)
[array([ 50.49203535, 51.65935211, 52.91505153, ..., 1998.0728519 ,
1999.24531165, 2000. ]),
array([0.50033912, 0.50019802, 0.50017659, ..., 0.50033534, 0.5002007 ,
0.50003559]),
array([0.49956204, 0.49974488, 0.49977235, ..., 0.4995669 , 0.49974139,
0.49995525]),
array([1.00071352, 1.00041571, 1.00037147, ..., 1.00070561, 1.00042142,
1.00007325])]
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))
for i,m in enumerate(np.logspace(0,2,mnum)):
print(i)
for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
sol = solve_original_model(m=m, sigma=sigma,tmax=1000,start_from=50)
oligomers = num_oligomers(sol)
final_oligomer_number = oligomers[-1][-1]
mat_oligomers[i,j] = final_oligomer_number
print(m,sigma,final_oligomer_number)
0 1.0 0.1 9497.23837149094 1.0 0.11844845812380056 8021.484175784347 1.0 0.14030037231905737 6772.82796507447 1.0 0.16618362775387496 5719.918675913305 1.0 0.19684194472866123 4827.278163999395 1.0 0.23315624847200295 4073.776007186317 1.0 0.27616998133438475 3440.050476115269 1.0 0.32711908469136664 2904.3435956801495 1.0 0.38746751204561314 2450.472386433967 1.0 0.45894929374867993 2069.556378595097 1.0 0.5436183620153837 1745.6864258858702 1.0 0.6439075678850823 1474.1702801649012 1.0 0.7626985859023444 1242.1469311128121 1.0 0.9034047151333576 1048.9131739112513 1.0 1.070068955693175 885.2087410258573 1.0 1.2674801788800205 747.7423336190142 1.0 1.5013107289081735 631.5121204366185 1.0 1.7782794100389228 533.1107262598389 1.0 2.106344542324121 450.28709037723485 1.0 2.4949326331577466 380.03100385401854 1.0 2.955209235202888 320.8941670323715 1.0 3.5003997734299794 271.12725753267574 1.0 4.146169559796819 228.87169427931437 1.0 4.911073914777701 193.2457600442409 1.0 5.817091329374358 163.17146862474078 1.0 6.890254987297223 137.73274182899087 1.0 8.16140079325183 116.33197385472813 1.0 9.66705340091042 98.20727783562079 1.0 11.450475699382825 82.93051826543011 1.0 13.562911913759416 70.02227422561164 1.0 16.065060038537286 59.108827143845176 1.0 19.028815912310247 49.907055988040725 1.0 22.539339047347912 42.13679158144729 1.0 26.697499572879334 35.57868997687477 1.0 31.622776601683793 30.037945080059576 1 1.1450475699382818 0.1 9472.6406926014 1.1450475699382818 0.11844845812380056 8020.104980880068 1.1450475699382818 0.14030037231905737 6773.13887241559 1.1450475699382818 0.16618362775387496 5718.923516827068 1.1450475699382818 0.19684194472866123 4827.614771476256 1.1450475699382818 0.23315624847200295 4074.178115851987 1.1450475699382818 0.27616998133438475 3441.0499962178296 1.1450475699382818 0.32711908469136664 2904.390166389578 1.1450475699382818 0.38746751204561314 2452.2301992232874 1.1450475699382818 0.45894929374867993 2069.499047267869 1.1450475699382818 0.5436183620153837 1746.5416406612144 1.1450475699382818 0.6439075678850823 1470.684629634692 1.1450475699382818 0.7626985859023444 1242.4369129919817 1.1450475699382818 0.9034047151333576 1048.8555412886092 1.1450475699382818 1.070068955693175 885.9942617453254 1.1450475699382818 1.2674801788800205 747.2102092624503 1.1450475699382818 1.5013107289081735 631.0077076753848 1.1450475699382818 1.7782794100389228 533.2623207342565 1.1450475699382818 2.106344542324121 450.21421250030494 1.1450475699382818 2.4949326331577466 379.93121486741904 1.1450475699382818 2.955209235202888 321.0456446950777 1.1450475699382818 3.5003997734299794 270.9569559674931 1.1450475699382818 4.146169559796819 228.75627640390118 1.1450475699382818 4.911073914777701 193.28319151314247 1.1450475699382818 5.817091329374358 163.17858232217498 1.1450475699382818 6.890254987297223 137.7422335772176 1.1450475699382818 8.16140079325183 116.34498272346269 1.1450475699382818 9.66705340091042 98.21010329274442 1.1450475699382818 11.450475699382825 82.93192602625192 1.1450475699382818 13.562911913759416 70.02336998880936 1.1450475699382818 16.065060038537286 59.12189137971139 1.1450475699382818 19.028815912310247 49.9061643648849 1.1450475699382818 22.539339047347912 42.13703020780156 1.1450475699382818 26.697499572879334 35.579007766609756 1.1450475699382818 31.622776601683793 30.037711172367935 2 1.3111339374215643 0.1 9468.236077233752 1.3111339374215643 0.11844845812380056 7999.906441799619 1.3111339374215643 0.14030037231905737 6759.704998723282 1.3111339374215643 0.16618362775387496 5720.736571917915 1.3111339374215643 0.19684194472866123 4828.0459865016255 1.3111339374215643 0.23315624847200295 4075.5023071728597 1.3111339374215643 0.27616998133438475 3440.0882637877357 1.3111339374215643 0.32711908469136664 2904.2888899491777 1.3111339374215643 0.38746751204561314 2451.9089093628145 1.3111339374215643 0.45894929374867993 2068.785214535877 1.3111339374215643 0.5436183620153837 1746.2095714260156 1.3111339374215643 0.6439075678850823 1472.054526470731 1.3111339374215643 0.7626985859023444 1242.1309726311356 1.3111339374215643 0.9034047151333576 1049.1965214410081 1.3111339374215643 1.070068955693175 885.9550991481425 1.3111339374215643 1.2674801788800205 747.9904740467043 1.3111339374215643 1.5013107289081735 631.3033048464517 1.3111339374215643 1.7782794100389228 532.8171429631162 1.3111339374215643 2.106344542324121 450.2622247284063 1.3111339374215643 2.4949326331577466 380.2413177728331 1.3111339374215643 2.955209235202888 320.95186810264346 1.3111339374215643 3.5003997734299794 271.09228770241964 1.3111339374215643 4.146169559796819 228.81949577334532 1.3111339374215643 4.911073914777701 193.19683164551347 1.3111339374215643 5.817091329374358 163.20267110498108 1.3111339374215643 6.890254987297223 137.75706391156834 1.3111339374215643 8.16140079325183 116.3492027963671 1.3111339374215643 9.66705340091042 98.20919774028899 1.3111339374215643 11.450475699382825 82.9309825445315 1.3111339374215643 13.562911913759416 70.0075905625565 1.3111339374215643 16.065060038537286 59.110795816576186 1.3111339374215643 19.028815912310247 49.90678742063192 1.3111339374215643 22.539339047347912 42.13730866547052 1.3111339374215643 26.697499572879334 35.5789186460052 1.3111339374215643 31.622776601683793 30.03785230962289 3 1.5013107289081733 0.1 9463.341115800968 1.5013107289081733 0.11844845812380056 7995.833620470002 1.5013107289081733 0.14030037231905737 6756.333521503364 1.5013107289081733 0.16618362775387496 5707.871263045132 1.5013107289081733 0.19684194472866123 4818.484811806256 1.5013107289081733 0.23315624847200295 4077.056050058158 1.5013107289081733 0.27616998133438475 3440.7050949374716 1.5013107289081733 0.32711908469136664 2904.715377964601 1.5013107289081733 0.38746751204561314 2451.0897200183886 1.5013107289081733 0.45894929374867993 2069.7945456951315 1.5013107289081733 0.5436183620153837 1746.9442293253953 1.5013107289081733 0.6439075678850823 1471.6940993709745 1.5013107289081733 0.7626985859023444 1242.0959065255386 1.5013107289081733 0.9034047151333576 1048.5871515040108 1.5013107289081733 1.070068955693175 885.2729205043015 1.5013107289081733 1.2674801788800205 747.7292598767892 1.5013107289081733 1.5013107289081735 631.1031558707759 1.5013107289081733 1.7782794100389228 532.9804196242517 1.5013107289081733 2.106344542324121 450.2837921778958 1.5013107289081733 2.4949326331577466 380.24403467467306 1.5013107289081733 2.955209235202888 320.89665984213684 1.5013107289081733 3.5003997734299794 270.9881102697371 1.5013107289081733 4.146169559796819 228.871004704445 1.5013107289081733 4.911073914777701 193.2265620336789 1.5013107289081733 5.817091329374358 163.11628222469255 1.5013107289081733 6.890254987297223 137.77028714824826 1.5013107289081733 8.16140079325183 116.31071251681253 1.5013107289081733 9.66705340091042 98.21344364690877 1.5013107289081733 11.450475699382825 82.93651627206394 1.5013107289081733 13.562911913759416 70.01111744023284 1.5013107289081733 16.065060038537286 59.111371471614994 1.5013107289081733 19.028815912310247 49.90727418632101 1.5013107289081733 22.539339047347912 42.137721539846176 1.5013107289081733 26.697499572879334 35.57441495913946 1.5013107289081733 31.622776601683793 30.034564838810827 4 1.7190722018585745 0.1 9458.630866084206 1.7190722018585745 0.11844845812380056 7989.108206255075 1.7190722018585745 0.14030037231905737 6753.043116233515 1.7190722018585745 0.16618362775387496 5708.307197868557 1.7190722018585745 0.19684194472866123 4821.09204428659 1.7190722018585745 0.23315624847200295 4067.282537101664 1.7190722018585745 0.27616998133438475 3440.8878391878716 1.7190722018585745 0.32711908469136664 2904.4438788081466 1.7190722018585745 0.38746751204561314 2452.367097891485 1.7190722018585745 0.45894929374867993 2069.7820788509002 1.7190722018585745 0.5436183620153837 1747.0852772413007 1.7190722018585745 0.6439075678850823 1471.9303241697555 1.7190722018585745 0.7626985859023444 1242.853047118037 1.7190722018585745 0.9034047151333576 1048.357322550013 1.7190722018585745 1.070068955693175 885.4231198469593 1.7190722018585745 1.2674801788800205 747.8205525553444 1.7190722018585745 1.5013107289081735 631.6555554933819 1.7190722018585745 1.7782794100389228 532.714109138598 1.7190722018585745 2.106344542324121 449.8586107939892 1.7190722018585745 2.4949326331577466 380.2353476012989 1.7190722018585745 2.955209235202888 320.79882191942556 1.7190722018585745 3.5003997734299794 270.936315352451 1.7190722018585745 4.146169559796819 228.80512448100345 1.7190722018585745 4.911073914777701 193.26385070953643 1.7190722018585745 5.817091329374358 163.1474176676202 1.7190722018585745 6.890254987297223 137.7847305063572 1.7190722018585745 8.16140079325183 116.31826119483542 1.7190722018585745 9.66705340091042 98.21799700250781 1.7190722018585745 11.450475699382825 82.93873409155113 1.7190722018585745 13.562911913759416 70.01303331483957 1.7190722018585745 16.065060038537286 59.114649385144624 1.7190722018585745 19.028815912310247 49.908327666370575 1.7190722018585745 22.539339047347912 42.13731484526492 1.7190722018585745 26.697499572879334 35.57419257016327 1.7190722018585745 31.622776601683793 30.034863440680137 5 1.9684194472866121 0.1 9457.597368276682 1.9684194472866121 0.11844845812380056 7982.009377178454 1.9684194472866121 0.14030037231905737 6748.091461880111 1.9684194472866121 0.16618362775387496 5706.583023966229 1.9684194472866121 0.19684194472866123 4821.0909595606145 1.9684194472866121 0.23315624847200295 4069.473777920445 1.9684194472866121 0.27616998133438475 3434.208202127424 1.9684194472866121 0.32711908469136664 2905.149817843093 1.9684194472866121 0.38746751204561314 2452.8145233435325 1.9684194472866121 0.45894929374867993 2070.0318093898613 1.9684194472866121 0.5436183620153837 1743.8553966964103 1.9684194472866121 0.6439075678850823 1471.479234879925 1.9684194472866121 0.7626985859023444 1242.943478185857 1.9684194472866121 0.9034047151333576 1048.667202560259 1.9684194472866121 1.070068955693175 885.2891767918519 1.9684194472866121 1.2674801788800205 748.0711038583133 1.9684194472866121 1.5013107289081735 631.4870810408984 1.9684194472866121 1.7782794100389228 533.2304626022726 1.9684194472866121 2.106344542324121 450.16767408830054 1.9684194472866121 2.4949326331577466 380.280370494065 1.9684194472866121 2.955209235202888 320.8129706133555 1.9684194472866121 3.5003997734299794 271.098978250728 1.9684194472866121 4.146169559796819 228.87637255764602 1.9684194472866121 4.911073914777701 193.29471158551445 1.9684194472866121 5.817091329374358 163.17403605018973 1.9684194472866121 6.890254987297223 137.79393539956868 1.9684194472866121 8.16140079325183 116.32716023873058 1.9684194472866121 9.66705340091042 98.2173352276246 1.9684194472866121 11.450475699382825 82.93878846486825 1.9684194472866121 13.562911913759416 70.01433575248265 1.9684194472866121 16.065060038537286 59.115720832458095 1.9684194472866121 19.028815912310247 49.90883971489732 1.9684194472866121 22.539339047347912 42.13773372076552 1.9684194472866121 26.697499572879334 35.57948685718625 1.9684194472866121 31.622776601683793 30.035397219367347 6 2.2539339047347906 0.1 9465.126643789701 2.2539339047347906 0.11844845812380056 7979.020462996187 2.2539339047347906 0.14030037231905737 6742.10060700211 2.2539339047347906 0.16618362775387496 5703.312500689324 2.2539339047347906 0.19684194472866123 4822.666005756447 2.2539339047347906 0.23315624847200295 4071.618405767408 2.2539339047347906 0.27616998133438475 3434.7306237368316 2.2539339047347906 0.32711908469136664 2898.3945425477373 2.2539339047347906 0.38746751204561314 2446.841522978335 2.2539339047347906 0.45894929374867993 2069.7664205097544 2.2539339047347906 0.5436183620153837 1744.0555797006648 2.2539339047347906 0.6439075678850823 1471.737171992609 2.2539339047347906 0.7626985859023444 1242.2260115254978 2.2539339047347906 0.9034047151333576 1048.527041735428 2.2539339047347906 1.070068955693175 886.2078346357296 2.2539339047347906 1.2674801788800205 747.7765765564113 2.2539339047347906 1.5013107289081735 631.6413976135017 2.2539339047347906 1.7782794100389228 533.150376146884 2.2539339047347906 2.106344542324121 449.95231441890076 2.2539339047347906 2.4949326331577466 380.00092739671237 2.2539339047347906 2.955209235202888 320.8161359174143 2.2539339047347906 3.5003997734299794 271.0347478364891 2.2539339047347906 4.146169559796819 228.7722848698983 2.2539339047347906 4.911073914777701 193.20967515133168 2.2539339047347906 5.817091329374358 163.2064647660301 2.2539339047347906 6.890254987297223 137.75371552014792 2.2539339047347906 8.16140079325183 116.33176921804179 2.2539339047347906 9.66705340091042 98.22803217314635 2.2539339047347906 11.450475699382825 82.93879251387823 2.2539339047347906 13.562911913759416 70.01394878038903 2.2539339047347906 16.065060038537286 59.116610325126 2.2539339047347906 19.028815912310247 49.911077580519475 2.2539339047347906 22.539339047347912 42.139095891177014 2.2539339047347906 26.697499572879334 35.5748796736127 2.2539339047347906 31.622776601683793 30.035549024779584 7 2.580861540418075 0.1 9490.032880311614 2.580861540418075 0.11844845812380056 7982.480459690292 2.580861540418075 0.14030037231905737 6735.788729391186 2.580861540418075 0.16618362775387496 5696.9402516913 2.580861540418075 0.19684194472866123 4821.393532660229 2.580861540418075 0.23315624847200295 4072.512316914074 2.580861540418075 0.27616998133438475 3436.117489648797 2.580861540418075 0.32711908469136664 2898.23616187275 2.580861540418075 0.38746751204561314 2447.236718068838 2.580861540418075 0.45894929374867993 2065.5472669486944 2.580861540418075 0.5436183620153837 1744.5991469530122 2.580861540418075 0.6439075678850823 1472.4684654094028 2.580861540418075 0.7626985859023444 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76.26985859023443 0.27616998133438475 106955.381708353 76.26985859023443 0.32711908469136664 85707.6847890131 76.26985859023443 0.38746751204561314 70917.69455269887 76.26985859023443 0.45894929374867993 55665.58306092853 76.26985859023443 0.5436183620153837 45059.04827986029 76.26985859023443 0.6439075678850823 37401.19690652643 76.26985859023443 0.7626985859023444 30326.809291278183 76.26985859023443 0.9034047151333576 24509.857153681547 76.26985859023443 1.070068955693175 19298.92798372344 76.26985859023443 1.2674801788800205 16008.16931106661 76.26985859023443 1.5013107289081735 12839.824244470175 76.26985859023443 1.7782794100389228 10139.380124637966 76.26985859023443 2.106344542324121 8039.923535818338 76.26985859023443 2.4949326331577466 6182.4399854187905 76.26985859023443 2.955209235202888 4699.420513679017 76.26985859023443 3.5003997734299794 3304.9556210456212 76.26985859023443 4.146169559796819 2072.7462368255165 76.26985859023443 4.911073914777701 858.5168133537856 76.26985859023443 5.817091329374358 163.1617447760726 76.26985859023443 6.890254987297223 137.7552945307636 76.26985859023443 8.16140079325183 116.30138007918528 76.26985859023443 9.66705340091042 98.21388649860914 76.26985859023443 11.450475699382825 82.93430460783574 76.26985859023443 13.562911913759416 70.01809581378053 76.26985859023443 16.065060038537286 59.11489101984434 76.26985859023443 19.028815912310247 49.90986781229777 76.26985859023443 22.539339047347912 42.138912428527235 76.26985859023443 26.697499572879334 35.57596077430674 76.26985859023443 31.622776601683793 30.0377625835097 33 87.33261623828433 0.1 586476.9901977802 87.33261623828433 0.11844845812380056 456445.9541731437 87.33261623828433 0.14030037231905737 346636.153486181 87.33261623828433 0.16618362775387496 253889.2054792302 87.33261623828433 0.19684194472866123 193839.10916512876 87.33261623828433 0.23315624847200295 165434.56568491852 87.33261623828433 0.27616998133438475 134589.45249835952 87.33261623828433 0.32711908469136664 100324.85694102883 87.33261623828433 0.38746751204561314 83986.24966897808 87.33261623828433 0.45894929374867993 67565.53961598183 87.33261623828433 0.5436183620153837 54791.46840069888 87.33261623828433 0.6439075678850823 42619.55593176349 87.33261623828433 0.7626985859023444 35811.286095909076 87.33261623828433 0.9034047151333576 29213.74510676359 87.33261623828433 1.070068955693175 23362.774309645356 87.33261623828433 1.2674801788800205 19072.149167622356 87.33261623828433 1.5013107289081735 15278.909758905893 87.33261623828433 1.7782794100389228 12283.213747295158 87.33261623828433 2.106344542324121 9851.69378829054 87.33261623828433 2.4949326331577466 7719.950424500689 87.33261623828433 2.955209235202888 5923.479812012361 87.33261623828433 3.5003997734299794 4451.560086226327 87.33261623828433 4.146169559796819 3059.740735388424 87.33261623828433 4.911073914777701 1777.7216635567547 87.33261623828433 5.817091329374358 163.56573657553653 87.33261623828433 6.890254987297223 137.7595296059756 87.33261623828433 8.16140079325183 116.31478468199533 87.33261623828433 9.66705340091042 98.23138123953139 87.33261623828433 11.450475699382825 82.92220532874191 87.33261623828433 13.562911913759416 70.02389225258807 87.33261623828433 16.065060038537286 59.11633197285232 87.33261623828433 19.028815912310247 49.91041077077265 87.33261623828433 22.539339047347912 42.140242562570144 87.33261623828433 26.697499572879334 35.57443940044335 87.33261623828433 31.622776601683793 30.03630185520499 34 100.0 0.1 705923.7676570666 100.0 0.11844845812380056 557279.2371946435 100.0 0.14030037231905737 431745.62433804997 100.0 0.16618362775387496 325728.42626543297 100.0 0.19684194472866123 236157.8072892566 100.0 0.23315624847200295 188323.00931877817 100.0 0.27616998133438475 160274.23495368764
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = plt.axes(111, projection='3d')
ax.set_yscale("log")
ax.set_xscale("log")
# ax.set_zscale("log")
x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
ax.plot_wireframe(x,y, np.log(mat_oligomers.T))
ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('ln(oligomers)')
ax.view_init(elev=30, azim=120)
plt.show()
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta*m/(1+P)
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
def num_oligomers(sol):
ts = sol[0]
Ps = sol[3]
cumulative_integral = [0] * len(ts)
for i in range(2, len(ts)):
area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
cumulative_integral[i] = cumulative_integral[i-1] + area
return([ts,cumulative_integral])
sol = solve(m=5000,beta=0.4,tmax=5000)
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([4500,5000])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"$\beta = 0.4$ and $m = 5000$")
plt.show()
oligomers = num_oligomers(sol)
plt.plot(oligomers[0],oligomers[1],color="red")
[<matplotlib.lines.Line2D at 0x7fe1d5a638e0>]
beta = 5
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))
for i,m in enumerate(np.logspace(0,2,mnum)):
print(i)
for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
oligomers = num_oligomers(sol)
final_oligomer_number = oligomers[-1][-1]
mat_oligomers[i,j] = final_oligomer_number
print(m,sigma,final_oligomer_number)
0 1.0 0.1 511.41766281010337 1.0 0.11844845812380056 508.282307598556 1.0 0.14030037231905737 504.6094489466907 1.0 0.16618362775387496 500.3149069386491 1.0 0.19684194472866123 495.30446272587585 1.0 0.23315624847200295 489.47253410520057 1.0 0.27616998133438475 482.7046971936214 1.0 0.32711908469136664 474.82948603786764 1.0 0.38746751204561314 465.9060063600262 1.0 0.45894929374867993 455.5689395363075 1.0 0.5436183620153837 443.7709516259087 1.0 0.6439075678850823 429.95827983023764 1.0 0.7626985859023444 415.05735626377776 1.0 0.9034047151333576 397.93644606713445 1.0 1.070068955693175 379.3205528228079 1.0 1.2674801788800205 359.0752990157088 1.0 1.5013107289081735 337.05017398350293 1.0 1.7782794100389228 313.80658829990904 1.0 2.106344542324121 289.4911600128019 1.0 2.4949326331577466 264.7078347305417 1.0 2.955209235202888 239.32259890734198 1.0 3.5003997734299794 214.5604295293716 1.0 4.146169559796819 190.46533457148684 1.0 4.911073914777701 167.68565500827373 1.0 5.817091329374358 146.39611443138693 1.0 6.890254987297223 126.91870854305198 1.0 8.16140079325183 109.3999702720464 1.0 9.66705340091042 93.86026440117685 1.0 11.450475699382825 80.18721126713496 1.0 13.562911913759416 68.32501650281323 1.0 16.065060038537286 58.07714643637219 1.0 19.028815912310247 49.2682788563647 1.0 22.539339047347912 41.74960100979572 1.0 26.697499572879334 35.340518737938105 1.0 31.622776601683793 29.891223943102787 1 1.1450475699382818 0.1 472.63868887726437 1.1450475699382818 0.11844845812380056 469.99960666938983 1.1450475699382818 0.14030037231905737 466.9029897820838 1.1450475699382818 0.16618362775387496 463.2755525575713 1.1450475699382818 0.19684194472866123 459.035570239308 1.1450475699382818 0.23315624847200295 454.0908235594041 1.1450475699382818 0.27616998133438475 448.3410275073302 1.1450475699382818 0.32711908469136664 441.58914820654854 1.1450475699382818 0.38746751204561314 433.8969072868071 1.1450475699382818 0.45894929374867993 425.1155464603425 1.1450475699382818 0.5436183620153837 414.5316522492409 1.1450475699382818 0.6439075678850823 403.08482376934745 1.1450475699382818 0.7626985859023444 390.163110231791 1.1450475699382818 0.9034047151333576 375.23705480869137 1.1450475699382818 1.070068955693175 358.90707067502365 1.1450475699382818 1.2674801788800205 341.0865479399444 1.1450475699382818 1.5013107289081735 321.4351051912131 1.1450475699382818 1.7782794100389228 300.5502666442933 1.1450475699382818 2.106344542324121 278.4995537248994 1.1450475699382818 2.4949326331577466 255.60337938750845 1.1450475699382818 2.955209235202888 232.37340588479088 1.1450475699382818 3.5003997734299794 209.26343380290646 1.1450475699382818 4.146169559796819 186.51587298984822 1.1450475699382818 4.911073914777701 164.84861738610914 1.1450475699382818 5.817091329374358 144.39894300387053 1.1450475699382818 6.890254987297223 125.60323562960279 1.1450475699382818 8.16140079325183 108.51752696763496 1.1450475699382818 9.66705340091042 93.28644904060269 1.1450475699382818 11.450475699382825 79.82022424129434 1.1450475699382818 13.562911913759416 68.09176938168913 1.1450475699382818 16.065060038537286 57.92986760944472 1.1450475699382818 19.028815912310247 49.18310216435592 1.1450475699382818 22.539339047347912 41.69150046738988 1.1450475699382818 26.697499572879334 35.30445543406245 1.1450475699382818 31.622776601683793 29.872058290294298 2 1.3111339374215643 0.1 437.00205505118583 1.3111339374215643 0.11844845812380056 434.7779241014333 1.3111339374215643 0.14030037231905737 432.1652068208496 1.3111339374215643 0.16618362775387496 429.10175332113323 1.3111339374215643 0.19684194472866123 425.5161997614192 1.3111339374215643 0.23315624847200295 421.3281872320487 1.3111339374215643 0.27616998133438475 416.44940063003645 1.3111339374215643 0.32711908469136664 410.7994126985323 1.3111339374215643 0.38746751204561314 404.26150956564646 1.3111339374215643 0.45894929374867993 396.65722570099297 1.3111339374215643 0.5436183620153837 387.59000904269146 1.3111339374215643 0.6439075678850823 377.7125793978103 1.3111339374215643 0.7626985859023444 366.14387999820235 1.3111339374215643 0.9034047151333576 353.5784551335615 1.3111339374215643 1.070068955693175 339.24588081707947 1.3111339374215643 1.2674801788800205 323.1769552043228 1.3111339374215643 1.5013107289081735 306.032146256447 1.3111339374215643 1.7782794100389228 287.41150989489796 1.3111339374215643 2.106344542324121 267.43960766448987 1.3111339374215643 2.4949326331577466 246.56270719312514 1.3111339374215643 2.955209235202888 225.19940013613495 1.3111339374215643 3.5003997734299794 203.69778502973054 1.3111339374215643 4.146169559796819 182.35165083117815 1.3111339374215643 4.911073914777701 161.81540503071346 1.3111339374215643 5.817091329374358 142.25932883535677 1.3111339374215643 6.890254987297223 124.10882352647442 1.3111339374215643 8.16140079325183 107.55205675711997 1.3111339374215643 9.66705340091042 92.64730507684008 1.3111339374215643 11.450475699382825 79.41074472507066 1.3111339374215643 13.562911913759416 67.82804595561943 1.3111339374215643 16.065060038537286 57.76185578895201 1.3111339374215643 19.028815912310247 49.078465498712596 1.3111339374215643 22.539339047347912 41.631787021014595 1.3111339374215643 26.697499572879334 35.26831563981717 1.3111339374215643 31.622776601683793 29.847383432197837 3 1.5013107289081733 0.1 404.4581683016634 1.5013107289081733 0.11844845812380056 402.579427442462 1.5013107289081733 0.14030037231905737 400.3712162787649 1.5013107289081733 0.16618362775387496 397.77905270320406 1.5013107289081733 0.19684194472866123 394.7410757923733 1.5013107289081733 0.23315624847200295 391.1875621833229 1.5013107289081733 0.27616998133438475 387.04144647120336 1.5013107289081733 0.32711908469136664 382.1278092075255 1.5013107289081733 0.38746751204561314 376.2007087161345 1.5013107289081733 0.45894929374867993 369.720739301583 1.5013107289081733 0.5436183620153837 362.2626530031501 1.5013107289081733 0.6439075678850823 353.3939759212841 1.5013107289081733 0.7626985859023444 343.7385958904093 1.5013107289081733 0.9034047151333576 332.4805069595212 1.5013107289081733 1.070068955693175 319.97618018584944 1.5013107289081733 1.2674801788800205 306.16112499857724 1.5013107289081733 1.5013107289081735 290.77453310831436 1.5013107289081733 1.7782794100389228 274.09743036942433 1.5013107289081733 2.106344542324121 256.21040655767115 1.5013107289081733 2.4949326331577466 237.4458217790953 1.5013107289081733 2.955209235202888 217.80949288210084 1.5013107289081733 3.5003997734299794 197.9159276101482 1.5013107289081733 4.146169559796819 177.99043578969213 1.5013107289081733 4.911073914777701 158.5755086308655 1.5013107289081733 5.817091329374358 139.93999792297643 1.5013107289081733 6.890254987297223 122.49820789071302 1.5013107289081733 8.16140079325183 106.44412061021633 1.5013107289081733 9.66705340091042 91.91046847194896 1.5013107289081733 11.450475699382825 78.94994703892694 1.5013107289081733 13.562911913759416 67.53521334080516 1.5013107289081733 16.065060038537286 57.57476926265253 1.5013107289081733 19.028815912310247 48.962389976208684 1.5013107289081733 22.539339047347912 41.558218997592704 1.5013107289081733 26.697499572879334 35.2222446908222 1.5013107289081733 31.622776601683793 29.822163847588484 4 1.7190722018585745 0.1 374.5511140806715 1.7190722018585745 0.11844845812380056 372.9566884676533 1.7190722018585745 0.14030037231905737 370.6964721369246 1.7190722018585745 0.16618362775387496 368.49801101013804 1.7190722018585745 0.19684194472866123 365.91937415425497 1.7190722018585745 0.23315624847200295 362.90019598482303 1.7190722018585745 0.27616998133438475 359.42390447127633 1.7190722018585745 0.32711908469136664 355.2767565087485 1.7190722018585745 0.38746751204561314 350.50736574954186 1.7190722018585745 0.45894929374867993 344.9643169298089 1.7190722018585745 0.5436183620153837 338.23643357172523 1.7190722018585745 0.6439075678850823 330.9283654361597 1.7190722018585745 0.7626985859023444 322.28375402369767 1.7190722018585745 0.9034047151333576 312.5359678710164 1.7190722018585745 1.070068955693175 301.60413706989107 1.7190722018585745 1.2674801788800205 289.490107401534 1.7190722018585745 1.5013107289081735 275.9044163415283 1.7190722018585745 1.7782794100389228 261.08830193019077 1.7190722018585745 2.106344542324121 245.05307778010578 1.7190722018585745 2.4949326331577466 228.09838480658033 1.7190722018585745 2.955209235202888 210.1865171365714 1.7190722018585745 3.5003997734299794 191.85229142767952 1.7190722018585745 4.146169559796819 173.4052666002215 1.7190722018585745 4.911073914777701 155.1698532640801 1.7190722018585745 5.817091329374358 137.454572589846 1.7190722018585745 6.890254987297223 120.80617306715513 1.7190722018585745 8.16140079325183 105.2832388446823 1.7190722018585745 9.66705340091042 91.12664512436503 1.7190722018585745 11.450475699382825 78.43711993112973 1.7190722018585745 13.562911913759416 67.208200909167 1.7190722018585745 16.065060038537286 57.36707873619565 1.7190722018585745 19.028815912310247 48.8299077947072 1.7190722018585745 22.539339047347912 41.47606829179049 1.7190722018585745 26.697499572879334 35.17048048790467 1.7190722018585745 31.622776601683793 29.790118805901052 5 1.9684194472866121 0.1 346.80244750775967 1.9684194472866121 0.11844845812380056 345.45516462696764 1.9684194472866121 0.14030037231905737 343.82046752169185 1.9684194472866121 0.16618362775387496 341.95538466470674 1.9684194472866121 0.19684194472866123 339.76565485382 1.9684194472866121 0.23315624847200295 337.19880371358164 1.9684194472866121 0.27616998133438475 333.8706464655501 1.9684194472866121 0.32711908469136664 330.4057373249132 1.9684194472866121 0.38746751204561314 326.3321180578828 1.9684194472866121 0.45894929374867993 321.6036607916845 1.9684194472866121 0.5436183620153837 315.82433538278343 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0.27616998133438475 56.851039739914185 58.170913293743574 0.32711908469136664 56.76550564299215 58.170913293743574 0.38746751204561314 56.65249422345473 58.170913293743574 0.45894929374867993 56.53297727322894 58.170913293743574 0.5436183620153837 56.38083905622133 58.170913293743574 0.6439075678850823 56.201864445453715 58.170913293743574 0.7626985859023444 56.00557028515014 58.170913293743574 0.9034047151333576 55.76317229675185 58.170913293743574 1.070068955693175 55.46630708982756 58.170913293743574 1.2674801788800205 55.134847364394304 58.170913293743574 1.5013107289081735 54.73357836286162 58.170913293743574 1.7782794100389228 54.26976673990238 58.170913293743574 2.106344542324121 53.716487516657224 58.170913293743574 2.4949326331577466 53.07812862020555 58.170913293743574 2.955209235202888 52.33810562096295 58.170913293743574 3.5003997734299794 51.46472323428797 58.170913293743574 4.146169559796819 50.46180293506366 58.170913293743574 4.911073914777701 49.30244619858274 58.170913293743574 5.817091329374358 47.96566539177826 58.170913293743574 6.890254987297223 46.44528413235304 58.170913293743574 8.16140079325183 44.71393392980091 58.170913293743574 9.66705340091042 42.77571890273784 58.170913293743574 11.450475699382825 40.62215887461365 58.170913293743574 13.562911913759416 38.25907063995463 58.170913293743574 16.065060038537286 35.70424843700839 58.170913293743574 19.028815912310247 32.990742902144255 58.170913293743574 22.539339047347912 30.162504118557674 58.170913293743574 26.697499572879334 27.281225765778363 58.170913293743574 31.622776601683793 24.41284582256461 31 66.60846290809158 0.1 53.32551236033137 66.60846290809158 0.11844845812380056 53.29844723689967 66.60846290809158 0.14030037231905737 53.266407670580875 66.60846290809158 0.16618362775387496 53.22848329159618 66.60846290809158 0.19684194472866123 53.18486438057473 66.60846290809158 0.23315624847200295 53.13174791258006 66.60846290809158 0.27616998133438475 53.0579724463283 66.60846290809158 0.32711908469136664 52.98366839184616 66.60846290809158 0.38746751204561314 52.884860363162105 66.60846290809158 0.45894929374867993 52.78102351264421 66.60846290809158 0.5436183620153837 52.64850170503291 66.60846290809158 0.6439075678850823 52.5035934447228 66.60846290809158 0.7626985859023444 52.32349995864913 66.60846290809158 0.9034047151333576 52.11266785526541 66.60846290809158 1.070068955693175 51.85399929438382 66.60846290809158 1.2674801788800205 51.56540276138465 66.60846290809158 1.5013107289081735 51.21687980251103 66.60846290809158 1.7782794100389228 50.809936433397695 66.60846290809158 2.106344542324121 50.33903472492882 66.60846290809158 2.4949326331577466 49.77055962336462 66.60846290809158 2.955209235202888 49.12364596281132 66.60846290809158 3.5003997734299794 48.358100285584094 66.60846290809158 4.146169559796819 47.47864778201853 66.60846290809158 4.911073914777701 46.459041700256286 66.60846290809158 5.817091329374358 45.28140765796982 66.60846290809158 6.890254987297223 43.93698698388254 66.60846290809158 8.16140079325183 42.41026627618278 66.60846290809158 9.66705340091042 40.68287590897409 66.60846290809158 11.450475699382825 38.755895728378675 66.60846290809158 13.562911913759416 36.628998223000686 66.60846290809158 16.065060038537286 34.313861982688685 66.60846290809158 19.028815912310247 31.8411646617076 66.60846290809158 22.539339047347912 29.240129054076306 66.60846290809158 26.697499572879334 26.562506126292426 66.60846290809158 31.622776601683793 23.873276554596085 32 76.26985859023443 0.1 49.75818012910519 76.26985859023443 0.11844845812380056 49.7346668223998 76.26985859023443 0.14030037231905737 49.70683054073009 76.26985859023443 0.16618362775387496 49.67388010686295 76.26985859023443 0.19684194472866123 49.63488019156462 76.26985859023443 0.23315624847200295 49.58050603038659 76.26985859023443 0.27616998133438475 49.525900234866086 76.26985859023443 0.32711908469136664 49.46131776550998 76.26985859023443 0.38746751204561314 49.375880897654135 76.26985859023443 0.45894929374867993 49.28556469419561 76.26985859023443 0.5436183620153837 49.169171047402564 76.26985859023443 0.6439075678850823 49.03508224985011 76.26985859023443 0.7626985859023444 48.88627370463298 76.26985859023443 0.9034047151333576 48.693858934013534 76.26985859023443 1.070068955693175 48.47751746264649 76.26985859023443 1.2674801788800205 48.22577064467799 76.26985859023443 1.5013107289081735 47.92249466282727 76.26985859023443 1.7782794100389228 47.56813253692192 76.26985859023443 2.106344542324121 47.148072349202984 76.26985859023443 2.4949326331577466 46.659793683265704 76.26985859023443 2.955209235202888 46.09481860080124 76.26985859023443 3.5003997734299794 45.42414380720736 76.26985859023443 4.146169559796819 44.652251585546615 76.26985859023443 4.911073914777701 43.75651580918252 76.26985859023443 5.817091329374358 42.71941843236362 76.26985859023443 6.890254987297223 41.53333460570119 76.26985859023443 8.16140079325183 40.17944832928727 76.26985859023443 9.66705340091042 38.643947130302436 76.26985859023443 11.450475699382825 36.92793057537601 76.26985859023443 13.562911913759416 35.01880312086227 76.26985859023443 16.065060038537286 32.93067793598568 76.26985859023443 19.028815912310247 30.679868131203655 76.26985859023443 22.539339047347912 28.293657710170837 76.26985859023443 26.697499572879334 25.8198562425443 76.26985859023443 31.622776601683793 23.306180366110635 33 87.33261623828433 0.1 46.43649451457636 87.33261623828433 0.11844845812380056 46.41605766701736 87.33261623828433 0.14030037231905737 46.383857094916216 87.33261623828433 0.16618362775387496 46.355225439287736 87.33261623828433 0.19684194472866123 46.32133536653159 87.33261623828433 0.23315624847200295 46.28120058842418 87.33261623828433 0.27616998133438475 46.22525245153999 87.33261623828433 0.32711908469136664 46.16912172603966 87.33261623828433 0.38746751204561314 46.10271752186752 87.33261623828433 0.45894929374867993 46.0170959491152 87.33261623828433 0.5436183620153837 45.924270645671314 87.33261623828433 0.6439075678850823 45.80754072645235 87.33261623828433 0.7626985859023444 45.67031240778621 87.33261623828433 0.9034047151333576 45.509302998894185 87.33261623828433 1.070068955693175 45.321309911961684 87.33261623828433 1.2674801788800205 45.09427056821317 87.33261623828433 1.5013107289081735 44.830825357680204 87.33261623828433 1.7782794100389228 44.52121132793289 87.33261623828433 2.106344542324121 44.162540679427536 87.33261623828433 2.4949326331577466 43.735948233764894 87.33261623828433 2.955209235202888 43.23437595431577 87.33261623828433 3.5003997734299794 42.654706279516596 87.33261623828433 4.146169559796819 41.977274414576165 87.33261623828433 4.911073914777701 41.185266230249866 87.33261623828433 5.817091329374358 40.2785264177356 87.33261623828433 6.890254987297223 39.23126535134751 87.33261623828433 8.16140079325183 38.03379164588084 87.33261623828433 9.66705340091042 36.67075170804738 87.33261623828433 11.450475699382825 35.14109228773759 87.33261623828433 13.562911913759416 33.43432341200013 87.33261623828433 16.065060038537286 31.552977886951673 87.33261623828433 19.028815912310247 29.51005638778611 87.33261623828433 22.539339047347912 27.330708531835093 87.33261623828433 26.697499572879334 25.051600948797024 87.33261623828433 31.622776601683793 22.713341670022366 34 100.0 0.1 43.335390087438796 100.0 0.11844845812380056 43.31761501643087 100.0 0.14030037231905737 43.29658364429708 100.0 0.16618362775387496 43.265373545534274 100.0 0.19684194472866123 43.23585652230511 100.0 0.23315624847200295 43.2009670535969 100.0 0.27616998133438475 43.15280492053543 100.0 0.32711908469136664 43.103982735327094 100.0 0.38746751204561314 43.04635911656055 100.0 0.45894929374867993 42.97154701458065 100.0 0.5436183620153837 42.88362423240803 100.0 0.6439075678850823 42.788229914034176 100.0 0.7626985859023444 42.668395086396714 100.0 0.9034047151333576 42.528939362792734 100.0 1.070068955693175 42.36605781206383 100.0 1.2674801788800205 42.168408397883574 100.0 1.5013107289081735 41.93740929051012 100.0 1.7782794100389228 41.668507898570965 100.0 2.106344542324121 41.3544078787129 100.0 2.4949326331577466 40.98266242652197 100.0 2.955209235202888 40.54387163643198 100.0 3.5003997734299794 40.03602421182966 100.0 4.146169559796819 39.44242886238599 100.0 4.911073914777701 38.752987685604786 100.0 5.817091329374358 37.94960941566005 100.0 6.890254987297223 37.028083292868246 100.0 8.16140079325183 35.97004586321035 100.0 9.66705340091042 34.76592498607281 100.0 11.450475699382825 33.400570637752075 100.0 13.562911913759416 31.873827681830967 100.0 16.065060038537286 30.18646749851656 100.0 19.028815912310247 28.340994999233388 100.0 22.539339047347912 26.35725125785444 100.0 26.697499572879334 24.262362817911903 100.0 31.622776601683793 22.09530063870254
np.savetxt('oligomers_a_beta-5.txt', mat_oligomers, fmt='%d')
import time
time.sleep(100)
from google.colab import files
# Download the file
files.download('oligomers_a_beta-5.txt')
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
X, Y = np.meshgrid(x, y) # Create a meshgrid for X and Y
ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))
ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('ln(oligomers)')
ax.view_init(elev=30, azim=120)
plt.show()
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta/P
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
def num_oligomers(sol):
ts = sol[0]
Ps = sol[3]
cumulative_integral = [0] * len(ts)
for i in range(2, len(ts)):
area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
cumulative_integral[i] = cumulative_integral[i-1] + area
return([ts,cumulative_integral])
sol = solve(m=200,beta=20)
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
import time
from google.colab import files
from mpl_toolkits.mplot3d import Axes3D
for beta in [20,10,5,1,0.5,0.1,0.05,0.01]:
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))
print(beta)
for i,m in enumerate(np.logspace(0,2,mnum)):
print(i)
for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
oligomers = num_oligomers(sol)
final_oligomer_number = oligomers[-1][-1]
mat_oligomers[i,j] = final_oligomer_number
np.savetxt('oligomers_b_beta-' + str(beta) + '.txt', mat_oligomers, fmt='%d')
time.sleep(20)
files.download('oligomers_b_beta-' + str(beta) + '.txt')
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
X, Y = np.meshgrid(x, y) # Create a meshgrid for X and Y
ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))
ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('oligomers')
ax.view_init(elev=30, azim=120)
plt.show()
20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
0.01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta*F
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
def num_oligomers(sol):
ts = sol[0]
Ps = sol[3]
cumulative_integral = [0] * len(ts)
for i in range(2, len(ts)):
area = 0.5 * (1/Ps[i] + 1/Ps[i-1]) * (ts[i] - ts[i-1])
cumulative_integral[i] = cumulative_integral[i-1] + area
return([ts,cumulative_integral])
sol = solve(m=5000000,beta=100)
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([50,200])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title(r"Solution with $\beta = 100$ and $m=5000000$")
plt.show()
import time
from google.colab import files
from mpl_toolkits.mplot3d import Axes3D
for beta in [0.005]:
mnum = 35
sigmanum = 35
mat_oligomers = np.zeros((mnum,sigmanum))
print(beta)
for i,m in enumerate(np.logspace(0,2,mnum)):
print(i)
for j,sigma in enumerate(np.logspace(-1,1.5,sigmanum)):
sol = solve(m=m,beta=beta, sigma=sigma,tmax=1000,start_from=50)
oligomers = num_oligomers(sol)
final_oligomer_number = oligomers[-1][-1]
mat_oligomers[i,j] = final_oligomer_number
np.savetxt('oligomers_c_beta-' + str(beta) + '.txt', mat_oligomers, fmt='%d')
time.sleep(20)
files.download('oligomers_c_beta-' + str(beta) + '.txt')
fig = plt.figure()
ax = fig.add_subplot(projection='3d')
x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
X, Y = np.meshgrid(x, y) # Create a meshgrid for X and Y
ax.plot_wireframe(X,Y , np.log(mat_oligomers.T))
ax.set_xlabel('m')
ax.set_ylabel('sigma')
ax.set_zlabel('oligomers')
ax.view_init(elev=30, azim=120)
plt.show()
0.005 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
import numpy as np
import matplotlib.pyplot as plt
import numpy as np
# Mount Google Drive
from google.colab import drive
drive.mount('/content/drive')
Mounted at /content/drive
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve_original_model(gamma=1,m=25,sigma=1,nu=1,
ic = [0,0,0], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
sol_m25 = solve_original_model(m=25)
plt.plot(sol_m25[0], sol_m25[1],label="F(t)",color="blue")
plt.plot(sol_m25[0],sol_m25[2],label="C(t)",color="green")
plt.plot(sol_m25[0],sol_m25[3],label="P(t)",color="red")
plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
from mpl_toolkits.mplot3d import Axes3D
mnum = 35
sigmanum = 35
def plot_matrix(file_path,title="",limi=14):
mat_oligomers = np.loadtxt(file_path, delimiter=' ')
fig = plt.figure()
ax = plt.axes(111, projection='3d')
x = np.logspace(0,2,mnum)
y = np.logspace(-1,1.5,sigmanum)
X, Y = np.meshgrid(x, y) # Create a meshgrid for X and Y
ax.plot_wireframe(X,Y,np.log(mat_oligomers.T),color="navy")
ax.set_xlabel('m')
ax.set_ylabel(r'$\sigma$')
ax.set_zlabel('ln (oligomer )')
ax.view_init(elev=30, azim=100)
plt.title(title)
ax.set_zlim([2,limi])
plt.show()
plot_matrix("/content/drive/MyDrive/Project/oligomers_normal.txt", "Original result")
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta*m/(1+P)
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
sol = solve(m=20000,beta=0.4,sigma=1,tmax=1000,ic=[2.5,250,90])
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([800,1000])
# plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution")
plt.show()
for beta in ["0.025","0.05","0.1","0.2","0.5","1","5"]:
plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_a_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0.1], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta/P
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
sol = solve(m=25,beta=0.1)
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
for beta in ["0.01","0.05","0.1","0.5","1","10","20"]:
plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_b_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
def solve(gamma=1,m=25,sigma=1,nu=1, beta = 1,
ic = [0,0,0], t0 = 0, tmax=200,start_from=50):
def derivs(t,var):
dvar = np.zeros(3)
F,C,P = var
dvar[0] = m/(1+P) - gamma * F*P
dvar[1] = gamma*F*P - nu*C
dvar[2] = sigma - P - gamma*F*P + nu*C + beta*F
return dvar
sol = solve_ivp(derivs,[t0,tmax],ic)
ind = list(sol.t>start_from).index(True)
return([sol.t[ind:], sol.y[0,:][ind:], sol.y[1,:][ind:], sol.y[2,:][ind:]])
sol = solve(m=25,beta=0.1)
plt.plot(sol[0], sol[1],label="F(t)",color="blue")
plt.plot(sol[0],sol[2],label="C(t)",color="green")
plt.plot(sol[0],sol[3],label="P(t)",color="red")
plt.xlim([50,200])
plt.ylim([0,220])
plt.legend()
plt.xlabel("Time")
plt.ylabel("Concentrations")
plt.title("Solution with m=25")
plt.show()
for beta in ["0.005","0.01","0.02","0.05 (1)","0.1","0.5","1","5","10"]:
plot_matrix("/content/drive/MyDrive/Project/oligomers"+"_c_beta-"+beta+".txt", "Protofibrils, beta = " + beta,limi=10)